The Wallabies' 1999 World Cup-winning full-back will not play again this season after rupturing his anterior cruciate knee ligament.
But the Falcons are keen to extend his stay on Tyneside into the 2008-09 campaign, with negotiations already under way.
Newcastle rugby director John Fletcher said: "We have held a number of meetings with Matt in the past fortnight to decide what is the best way ahead for both parties.
"His current contract with the club is up at the end of this season, but there is a massive determination on our part to retain his services.
"Before suffering his injury, Matt was playing some of the best rugby of his career, and even after what was a devastating blow he has remained an extremely positive presence around the team.
"He is a player who still falls firmly into the world-class bracket, and clearly when there is the opportunity to keep such a talent at the club we are keen to make it happen.
"There is nothing concrete in place yet for next season, but what we have seen during our initial meetings with Matt is a commitment from him to look seriously at the option of staying for another season."
Burke will shortly fly home to Australia, where he will begin the initial phase of his rehabilitation.
Fletcher added: "We will then sit down and look at next season in more detail with all the relevant information, before hopefully reaching a positive conclusion.
"Matt is a truly outstanding individual and remains very much part of this club. Our talks continue to progress well."
Burke joined Newcastle in 2004, and has amassed 430 points in 81 games.